{"id":4245,"date":"2022-11-24T14:59:21","date_gmt":"2022-11-24T05:59:21","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?p=4245"},"modified":"2023-11-21T17:36:43","modified_gmt":"2023-11-21T08:36:43","slug":"%e4%b8%8b%e3%81%93%e3%82%99%e3%81%97%e3%82%89%e3%81%88%e3%81%97%e3%81%9f%e4%b8%87%e6%9c%89%e5%bc%95%e5%8a%9b%e3%81%ae2%e4%bd%93%e5%95%8f%e9%a1%8c%e3%81%ae%e9%81%8b%e5%8b%95%e6%96%b9%e7%a8%8b%e5%bc%8f","status":"publish","type":"post","link":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/4245\/","title":{"rendered":"\u4e0b\u3053\u3099\u3057\u3089\u3048\u3057\u305f\u4e07\u6709\u5f15\u529b\u306e2\u4f53\u554f\u984c\u306e\u904b\u52d5\u65b9\u7a0b\u5f0f\u3092 Maxima \u3066\u3099\u6570\u5024\u7684\u306b\u89e3\u304f"},"content":{"rendered":"

\u300c\u4e07\u6709\u5f15\u529b\u306e2\u4f53\u554f\u984c\u306e\u904b\u52d5\u65b9\u7a0b\u5f0f\u3092\u6570\u5024\u7684\u306b\u89e3\u304f\u524d\u306e\u4e0b\u3054\u3057\u3089\u3048<\/a>\u300d\u3067\u4e0b\u3054\u3057\u3089\u3048\u3057\u305f\u904b\u52d5\u65b9\u7a0b\u5f0f\u3092 Maxima \u3067 Runge-Kutta \u6cd5\u3092\u4f7f\u3063\u3066\u6570\u5024\u7684\u306b\u89e3\u304f\u3068\u3044\u3046\u8a71\u3002<\/p>\n

<\/p>\n

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\u7121\u6b21\u5143\u5316\u3055\u308c\u305f\u904b\u52d5\u65b9\u7a0b\u5f0f<\/h3>\n

\u300c\u4e07\u6709\u5f15\u529b\u306e2\u4f53\u554f\u984c\u306e\u904b\u52d5\u65b9\u7a0b\u5f0f\u3092\u6570\u5024\u7684\u306b\u89e3\u304f\u524d\u306e\u4e0b\u3054\u3057\u3089\u3048<\/a>\u300d\u306b\u307e\u3068\u3081\u305f\u3088\u3046\u306b\uff0c\u7cfb\u306b\u7279\u5fb4\u7684\u306a\u9577\u3055\u3068\u6642\u9593\u30b9\u30b1\u30fc\u30eb\u3067\u7121\u6b21\u5143\u5316\u3057\u305f\u5909\u6570\u306b\u5bfe\u3059\u308b\u904b\u52d5\u65b9\u7a0b\u5f0f\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u306a\u308b\u306e\u3067\u3042\u3063\u305f\u3002<\/p>\n

\\begin{eqnarray}
\n\\frac{d^2 X}{dT^2} = – \\frac{X}{\\left(X^2 + Y^2\\right)^{\\frac{3}{2}}} \\\\
\n\\frac{d^2 Y}{dT^2} = – \\frac{Y}{\\left(X^2 + Y^2\\right)^{\\frac{3}{2}}}
\n\\end{eqnarray}<\/p>\n

\u521d\u671f\u6761\u4ef6<\/h3>\n

\u7121\u6b21\u5143\u5316\u3055\u308c\u305f\u5909\u6570\u306b\u5bfe\u3059\u308b\u521d\u671f\u6761\u4ef6\u306f $T = 0$ \u3067<\/p>\n

\\begin{eqnarray}
\nX &=& 1 \\\\
\nY &=& 0\\\\
\nV_x &=& \\frac{dX}{dT} = 0 \\\\
\nV_y &=& \\frac{dY}{dT} = \\frac{v_{y0}}{v_0} = k \\ \\ ( 0 < k < \\sqrt{2})
\n\\end{eqnarray}<\/p>\n

\u6955\u5186\u306e\u8ecc\u9053\u8981\u7d20\uff08\u9577\u534a\u5f84\uff0c\u96e2\u5fc3\u7387\uff0c\u5468\u671f\uff09<\/h3>\n

\u3053\u306e\u3068\u304d\uff0c\u6570\u5024\u8a08\u7b97\u3059\u308b\u524d\u306b\u3082\u3046\uff0c\u8ecc\u9053\u306f\u898f\u683c\u5316\u3055\u308c\u305f\u9577\u534a\u5f84\u304c<\/p>\n

$$ A \\equiv \\frac{a}{x_0} = \\frac{1}{2 – k^2} $$<\/p>\n

\u96e2\u5fc3\u7387\u304c<\/p>\n

$$e = |k^2 – 1|$$<\/p>\n

\u306e\u6955\u5186\u306b\u306a\u308b\u3053\u3068\u304c\u308f\u304b\u3063\u3066\u3057\u307e\u3046\u3002\u307e\u305f\uff0c\u898f\u683c\u5316\u3055\u308c\u305f\u5468\u671f\u304c<\/p>\n

$$P \\equiv \\frac{p}{\\tau} = 2 \\pi A^{\\frac{3}{2}} = \\frac{2 \\pi}{\\left(2 – k^2\\right)^{\\frac{3}{2}}}$$<\/p>\n

\u3068\u306a\u308b\u3053\u3068\u3082\u308f\u304b\u3063\u3066\u3057\u307e\u3063\u3066\u3044\u308b\u306e\u3067\u3042\u3063\u305f\u3002<\/p>\n

\u3058\u3083\u3042\uff0c\u305d\u3053\u307e\u3067\u308f\u304b\u3063\u3066\u3044\u308b\u306e\u306a\u3089\uff0c\u306a\u305c\u308f\u3056\u308f\u3056\u6570\u5024\u8a08\u7b97\u3059\u308b\u306e\u304b<\/strong><\/span>\u3068\u3044\u3046\u3068\uff0c\u305d\u308c\u306f\u6955\u5186\u8ecc\u9053\u306e\u89e3\u304c\u6642\u9593 $t$ \u306e\u967d\u95a2\u6570\u3068\u3057\u3066\u4e0e\u3048\u3089\u308c\u3066\u3044\u308b\u308f\u3051\u3067\u306f\u306a\u3044\u304b\u3089<\/strong><\/span>\u3002\u6642\u523b $t$ \u306e\u3068\u304d\uff0c\u3069\u3053\u306b\u3044\u308b\u304b\u304c\u89e3\u6790\u7684\u306b\u308f\u304b\u3063\u3066\u3044\u306a\u3044\u306e\u3067\uff0c\u305d\u308c\u3092\u77e5\u308a\u305f\u3044\u304b\u3089\u6570\u5024\u8a08\u7b97\u3059\u308b\u3053\u3068\u306b\u306a\u308b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n

 <\/p>\n

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Runge-Kutta \u7528\u306b\u9023\u7acb1\u968e\u5fae\u5206\u65b9\u7a0b\u5f0f\u7cfb\u306b<\/h3>\n

\u5168\u3066\u306e\u5909\u6570\u306f\u7121\u6b21\u5143\u5316\u3055\u308c\u3044\u308b\u3053\u3068\u5fd8\u308c\u306a\u3044\u3088\u3046\u306b\u3057\u3066\uff0cMaxima \u306e\u30b3\u30fc\u30c9\u306b\u3059\u308b\u969b\u306b\u306f\uff0c\uff08\u898b\u3084\u3059\u3055\u306e\u89b3\u70b9\u304b\u3089\uff09\u5c0f\u6587\u5b57\u3067\u66f8\u3044\u3066\u307f\u307e\u3059\u3002<\/p>\n

\\begin{eqnarray}
\n\\frac{dx}{dt} &=& F_1(x, y, v, w, t) \\Rightarrow F_1(v) = v \\\\
\n\\frac{dy}{dt} &=& F_2(x, y, v, w, t) \\Rightarrow F_2(w) = w \\\\
\n\\frac{dv}{dt} &=& F_3(x, y, v, w, t) \\Rightarrow F_3(x, y)
\n= -\\frac{x}{\\left(x^2 + y^2\\right)^{\\frac{3}{2}}} \\\\
\n\\frac{dw}{dt} &=& F_4(x, y, v, w, t) \\Rightarrow F_4(x, y)
\n= -\\frac{y}{\\left(x^2 + y^2\\right)^{\\frac{3}{2}}}
\n\\end{eqnarray}<\/p>\n<\/div>\n<\/div>\n<\/div>\n

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In\u00a0[1]:<\/div>\n
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F1<\/span>(<\/span>v<\/span>)<\/span>:=<\/span> v<\/span>;\r\nF2<\/span>(<\/span>w<\/span>)<\/span>:=<\/span> w<\/span>;\r\nF3<\/span>(<\/span>x<\/span>, y<\/span>)<\/span>:=<\/span> -<\/span>x<\/span>\/<\/span>(<\/span>x<\/span>**<\/span>2<\/span> +<\/span> y<\/span>**<\/span>2<\/span>)<\/span>**<\/span>(<\/span>3\/<\/span>2<\/span>)<\/span>;\r\nF4<\/span>(<\/span>x<\/span>, y<\/span>)<\/span>:=<\/span> -<\/span>y<\/span>\/<\/span>(<\/span>x<\/span>**<\/span>2<\/span> +<\/span> y<\/span>**<\/span>2<\/span>)<\/span>**<\/span>(<\/span>3\/<\/span>2<\/span>)<\/span>;\r\n\r\nP<\/span>(<\/span>k<\/span>)<\/span>:=<\/span> float<\/span>(<\/span>2*<\/span>%pi<\/span>\/<\/span>(<\/span>2<\/span> -<\/span> k<\/span>**<\/span>2<\/span>)<\/span>**<\/span>(<\/span>3\/<\/span>2<\/span>))<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[1]:<\/div>\n
\\[\\tag{${\\it \\%o}_{1}$}F_{1}\\left(v\\right):=v\\]<\/div>\n<\/div>\n
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\\[\\tag{${\\it \\%o}_{2}$}F_{2}\\left(w\\right):=w\\]<\/div>\n<\/div>\n
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\\[\\tag{${\\it \\%o}_{3}$}F_{3}\\left(x , y\\right):=\\frac{-x}{\\left(x^2+y^2\\right)^{\\frac{3}{2}}}\\]<\/div>\n<\/div>\n
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\\[\\tag{${\\it \\%o}_{4}$}F_{4}\\left(x , y\\right):=\\frac{-y}{\\left(x^2+y^2\\right)^{\\frac{3}{2}}}\\]<\/div>\n<\/div>\n
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Out[1]:<\/div>\n
\\[\\tag{${\\it \\%o}_{5}$}P\\left(k\\right):={\\it float}\\left(\\frac{2\\,\\pi}{\\left(2-k^2\\right)^{\\frac{3}{2}}}\\right)\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Runge-Kutta \u6cd5\u3067\u89e3\u304f<\/h3>\n
\u521d\u671f\u6761\u4ef6 $k = 1$ \u3067\u5186\u8ecc\u9053\u3068\u306a\u308b\u3053\u3068\u3092\u78ba\u8a8d<\/h4>\n
\u307e\u305a\u306f\uff0c$k = 1$ \u3068\u3044\u3046\u521d\u671f\u6761\u4ef6\u3067\u5186\u8ecc\u9053\u306b\u306a\u308b\u3053\u3068\u3092\u78ba\u8a8d\u30021\u5468\u671f\u3092 $N$ \u7b49\u5206\u3057\uff0c$t = P(1) = 2 \\pi$ \u307e\u3067\u8a08\u7b97\u3057\u305f\u3068\u304d\u306e\u7b54\u3048\u304c $x_0 = 1, y_0 = 0$ \u306b\u306a\u3063\u3066\u3044\u308b\u304b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[2]:<\/div>\n
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fpprintprec<\/span>:<\/span> 5$ \/* \u6d6e\u52d5\u5c0f\u6570\u70b9\u8868\u793a\u306f\u6709\u52b9\u6570\u5b57 5 \u6841\u306b *\/<\/span>\r\n\r\nN<\/span>:<\/span> 36*<\/span>20$\r\nh<\/span>:<\/span> P<\/span>(<\/span>1<\/span>)<\/span>\/<\/span>N<\/span>$\r\n\r\nsolc<\/span>:<\/span>\r\nrk<\/span>([<\/span>F1<\/span>(<\/span>v<\/span>)<\/span>, F2<\/span>(<\/span>w<\/span>)<\/span>, F3<\/span>(<\/span>x<\/span>, y<\/span>)<\/span>, F4<\/span>(<\/span>x<\/span>, y<\/span>)]<\/span>, \r\n   [<\/span>x<\/span>, y<\/span>, v<\/span>, w<\/span>]<\/span>, [<\/span>1<\/span>, 0<\/span>, 0<\/span>, 1]<\/span>, \r\n   [<\/span>t<\/span>, 0<\/span>, P<\/span>(<\/span>1<\/span>)<\/span>, h<\/span>])<\/span>$\r\n\r\nprint<\/span>(<\/span>\"N = \"<\/span>, N<\/span>, \"  h = \"<\/span>, h<\/span>)<\/span>$\r\nprint<\/span>(<\/span>\"x \u306e\u8aa4\u5dee \"<\/span>, 1<\/span> -<\/span> solc<\/span>[<\/span>length<\/span>(<\/span>solc<\/span>)][<\/span>2])<\/span>$\r\nprint<\/span>(<\/span>\"y \u306e\u8aa4\u5dee \"<\/span>, 0<\/span> -<\/span> solc<\/span>[<\/span>length<\/span>(<\/span>solc<\/span>)][<\/span>3])<\/span>$\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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N = \\(720\\) h = \\(0.0087266\\)<\/div>\n<\/div>\n
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x \u306e\u8aa4\u5dee \\(8.8332 \\times 10^{-12}\\)<\/div>\n<\/div>\n
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y \u306e\u8aa4\u5dee \\(-8.7663 \\times 10^{-10}\\)<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u3061\u306a\u307f\u306b\uff0cMaxima \u306e rk()<\/code> \u306f4\u6b21\u306e Runge-Kutta \u6cd5\u3067\u89e3\u304f\u306e\u3067\uff0c\u8aa4\u5dee\u306f $h^4$ \u306e\u7a0b\u5ea6\u3067\u3042\u308b\u3053\u3068\u304c\u4e88\u60f3\u3055\u308c\u308b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[3]:<\/div>\n
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\/* 4\u6b21\u306e Runge-Kutta \u6cd5\u306e\u8aa4\u5dee\u306f h**4 \u306e\u7a0b\u5ea6 *\/<\/span>\r\nh<\/span>**<\/span>4<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[3]:<\/div>\n
\\[\\tag{${\\it \\%o}_{13}$}5.7995 \\times 10^{-9}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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<\/div>\n
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$e = 0.6$ \u3068\u306a\u308b\u521d\u671f\u6761\u4ef6\u3067\u6570\u5024\u8a08\u7b97<\/h4>\n
\u521d\u671f\u6761\u4ef6 $k$ \u3068\u6955\u5186\u306e\u96e2\u5fc3\u7387 $e$ \u3068\u306e\u95a2\u4fc2\u306f $e = |k^2 – 1|$ \u3067\u3042\u3063\u305f\u304b\u3089…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[4]:<\/div>\n
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k<\/span>:<\/span> sqrt<\/span>(<\/span>1<\/span> +<\/span> 0.6<\/span>)<\/span>$\r\n\r\nN<\/span>:<\/span> 36*<\/span>80$\r\nh<\/span>:<\/span> P<\/span>(<\/span>k<\/span>)<\/span>\/<\/span>N<\/span>$\r\n\r\nsole<\/span>:<\/span>\r\nrk<\/span>([<\/span>F1<\/span>(<\/span>v<\/span>)<\/span>, F2<\/span>(<\/span>w<\/span>)<\/span>, F3<\/span>(<\/span>x<\/span>, y<\/span>)<\/span>, F4<\/span>(<\/span>x<\/span>, y<\/span>)]<\/span>, \r\n   [<\/span>x<\/span>, y<\/span>, v<\/span>, w<\/span>]<\/span>, [<\/span>1<\/span>, 0<\/span>, 0<\/span>, k<\/span>]<\/span>, \r\n   [<\/span>t<\/span>, 0<\/span>, P<\/span>(<\/span>k<\/span>)<\/span>, h<\/span>])<\/span>$\r\n\r\nprint<\/span>(<\/span>\"N = \"<\/span>, N<\/span>, \"  h = \"<\/span>, h<\/span>)<\/span>$\r\nprint<\/span>(<\/span>\"x \u306e\u8aa4\u5dee \"<\/span>, 1<\/span> -<\/span> sole<\/span>[<\/span>length<\/span>(<\/span>sole<\/span>)][<\/span>2])<\/span>$\r\nprint<\/span>(<\/span>\"y \u306e\u8aa4\u5dee \"<\/span>, 0<\/span> -<\/span> sole<\/span>[<\/span>length<\/span>(<\/span>sole<\/span>)][<\/span>3])<\/span>$\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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N = \\(2880\\) h = \\(0.0086238\\)<\/div>\n<\/div>\n
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x \u306e\u8aa4\u5dee \\(-4.0923 \\times 10^{-13}\\)<\/div>\n<\/div>\n
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y \u306e\u8aa4\u5dee \\(-7.0832 \\times 10^{-9}\\)<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[5]:<\/div>\n
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\/* 4\u6b21\u306e Runge-Kutta \u6cd5\u306e\u8aa4\u5dee\u306f h**4 \u306e\u7a0b\u5ea6 *\/<\/span>\r\nh<\/span>**<\/span>4<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[5]:<\/div>\n
\\[\\tag{${\\it \\%o}_{21}$}5.5308 \\times 10^{-9}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u30b1\u30d7\u30e9\u30fc\u65b9\u7a0b\u5f0f\u3092\u6570\u5024\u7684\u306b\u89e3\u3044\u305f\u5834\u5408\u306e\u89e3\u3068\u306e\u6bd4\u8f03<\/h3>\n
\u300c\u30b1\u30d7\u30e9\u30fc\u65b9\u7a0b\u5f0f\u3092\u6570\u5024\u7684\u306b\u89e3\u3044\u3066\u30b1\u30d7\u30e9\u30fc\u306e\u7b2c2\u6cd5\u5247\u3092\u8996\u899a\u7684\u306b\u78ba\u8a8d\u3059\u308b<\/a>\u300d\u306b\u307e\u3068\u3081\u3066\u3044\u308b\u3088\u3046\u306b<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[6]:<\/div>\n
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\/* \u96e2\u5fc3\u7387 *\/<\/span>\r\ne<\/span>:<\/span> 0.<\/span>6$\r\n\r\n\/* \u5206\u5272\u6570 *\/<\/span>\r\nN<\/span>:<\/span> 36$\r\n\r\n\/* \u30b1\u30d7\u30e9\u30fc\u65b9\u7a0b\u5f0f\u3092 find_root \u3067\u6570\u5024\u7684\u306b\u89e3\u304f *\/<\/span>\r\ng<\/span>(<\/span>u<\/span>)<\/span>:=<\/span> u<\/span> -<\/span> e<\/span>*<\/span>sin<\/span>(<\/span>u<\/span>)<\/span>$\r\nfor<\/span> i<\/span>:<\/span>0<\/span> thru<\/span> N<\/span> do<\/span> \r\n    u<\/span>[<\/span>i<\/span>]<\/span>:<\/span> find_root<\/span>(<\/span>g<\/span>(<\/span>u<\/span>)<\/span> =<\/span> 2*<\/span>%pi<\/span>\/<\/span>N<\/span> *<\/span> i<\/span>, u<\/span>, 0<\/span>, 2*<\/span>%pi<\/span>)<\/span>$\r\n\r\n\/* t = 0 \u3067 xu = 1 ==> a = 1\/(1-e) *\/<\/span>\r\nxu<\/span>(<\/span>u<\/span>)<\/span>:=<\/span> (<\/span>cos<\/span>(<\/span>u<\/span>)<\/span> -<\/span> e<\/span>)<\/span>\/<\/span>(<\/span>1-<\/span>e<\/span>)<\/span>$\r\nyu<\/span>(<\/span>u<\/span>)<\/span>:=<\/span> sqrt<\/span>(<\/span>1-<\/span>e<\/span>**<\/span>2<\/span>)<\/span>*<\/span>sin<\/span>(<\/span>u<\/span>)<\/span>\/<\/span>(<\/span>1-<\/span>e<\/span>)<\/span>$\r\n\r\nxy_kep<\/span>:<\/span> makelist<\/span>([<\/span>xu<\/span>(<\/span>u<\/span>[<\/span>i<\/span>])<\/span>, yu<\/span>(<\/span>u<\/span>[<\/span>i<\/span>])]<\/span>, i<\/span>, 0<\/span>, N<\/span>)<\/span>$\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[7]:<\/div>\n
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\/* Runge-Kutta \u6cd5\u3067\u6c42\u3081\u305f\u89e3\u3092\u9593\u5f15\u304f *\/<\/span>\r\nxy_rk<\/span>:<\/span> makelist<\/span>([<\/span>sole<\/span>[<\/span>i<\/span>][<\/span>2]<\/span>, sole<\/span>[<\/span>i<\/span>][<\/span>3]]<\/span>, i<\/span>, 1<\/span>, length<\/span>(<\/span>sole<\/span>)<\/span>, 80<\/span>)<\/span>$\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[8]:<\/div>\n
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\/* Runge-Kutta \u6cd5\u306e\u89e3\u3068\uff0c\u30b1\u30d7\u30e9\u30fc\u65b9\u7a0b\u5f0f\u306e\u6570\u5024\u89e3\u3092\u4f7f\u3063\u305f\u89e3\u3068\u306e\u5dee\u3092 *\/<\/span>\r\n\/* \u884c\u5217\u7684\u8868\u793a\u3067 *\/<\/span>\r\napply<\/span>(<\/span>'<\/span>matrix<\/span>, xy_rk<\/span> -<\/span> xy_kep<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[8]:<\/div>\n
\\[\\tag{${\\it \\%o}_{30}$}\\begin{pmatrix}0.0 & 0.0 \\\\ -5.0185 \\times 10^{-12} & -5.0332 \\times 10^{-11} \\\\ -6.3828 \\times 10^{-11} & -9.0501 \\times 10^{-11} \\\\ -1.3044 \\times 10^{-10} & -1.6713 \\times 10^{-10} \\\\ -1.8026 \\times 10^{-10} & -2.6871 \\times 10^{-10} \\\\ -2.0915 \\times 10^{-10} & -3.8353 \\times 10^{-10} \\\\ -2.1784 \\times 10^{-10} & -5.0432 \\times 10^{-10} \\\\ -2.0805 \\times 10^{-10} & -6.2665 \\times 10^{-10} \\\\ -1.815 \\times 10^{-10} & -7.4776 \\times 10^{-10} \\\\ -1.3965 \\times 10^{-10} & -8.6582 \\times 10^{-10} \\\\ -8.3679 \\times 10^{-11} & -9.7954 \\times 10^{-10} \\\\ -1.449 \\times 10^{-11} & -1.0879 \\times 10^{-9} \\\\ 6.7232 \\times 10^{-11} & -1.1902 \\times 10^{-9} \\\\ 1.6099 \\times 10^{-10} & -1.2856 \\times 10^{-9} \\\\ 2.6644 \\times 10^{-10} & -1.3734 \\times 10^{-9} \\\\ 3.834 \\times 10^{-10} & -1.453 \\times 10^{-9} \\\\ 5.1179 \\times 10^{-10} & -1.5235 \\times 10^{-9} \\\\ 6.5166 \\times 10^{-10} & -1.584 \\times 10^{-9} \\\\ 8.0321 \\times 10^{-10} & -1.6335 \\times 10^{-9} \\\\ 9.667 \\times 10^{-10} & -1.6709 \\times 10^{-9} \\\\ 1.1426 \\times 10^{-9} & -1.6947 \\times 10^{-9} \\\\ 1.3314 \\times 10^{-9} & -1.7031 \\times 10^{-9} \\\\ 1.5338 \\times 10^{-9} & -1.6939 \\times 10^{-9} \\\\ 1.7507 \\times 10^{-9} & -1.6645 \\times 10^{-9} \\\\ 1.9831 \\times 10^{-9} & -1.6111 \\times 10^{-9} \\\\ 2.2322 \\times 10^{-9} & -1.5293 \\times 10^{-9} \\\\ 2.4994 \\times 10^{-9} & -1.4128 \\times 10^{-9} \\\\ 2.7859 \\times 10^{-9} & -1.2529 \\times 10^{-9} \\\\ 3.0931 \\times 10^{-9} & -1.0375 \\times 10^{-9} \\\\ 3.4213 \\times 10^{-9} & -7.4884 \\times 10^{-10} \\\\ 3.7681 \\times 10^{-9} & -3.5962 \\times 10^{-10} \\\\ 4.1247 \\times 10^{-9} & 1.7348 \\times 10^{-10} \\\\ 4.4627 \\times 10^{-9} & 9.225 \\times 10^{-10} \\\\ 4.6974 \\times 10^{-9} & 2.0094 \\times 10^{-9} \\\\ 4.5688 \\times 10^{-9} & 3.6173 \\times 10^{-9} \\\\ 3.3243 \\times 10^{-9} & 5.7729 \\times 10^{-9} \\\\ 4.0923 \\times 10^{-13} & 7.0832 \\times 10^{-9} \\\\ \\end{pmatrix}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u4e00\u5b9a\u6642\u9593\u9593\u9694\u3054\u3068\u306e\u4f4d\u7f6e\u3092\u30b0\u30e9\u30d5\u306b<\/h3>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[9]:<\/div>\n
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draw2d<\/span>(<\/span>\r\n  title<\/span> =<\/span> \"\u4e00\u5b9a\u6642\u9593\u9593\u9694\u3054\u3068\u306e\u4f4d\u7f6e\"<\/span>,\r\n  \/* \u6ed1\u3089\u304b\u306b\u66f2\u7dda\u3092\u63cf\u304f\u3088\u3046\u306b *\/<\/span>\r\n  nticks<\/span> =<\/span> 100<\/span>, \r\n  \/* \u8868\u793a\u7bc4\u56f2 *\/<\/span>\r\n  xrange<\/span> =<\/span> [<\/span>-<\/span>4.5<\/span>, 1.<\/span>5]<\/span>, yrange<\/span> =<\/span> [<\/span>-<\/span>3<\/span>, 3]<\/span>,\r\n  \/* \u7e26\u6a2a\u6bd4\u3002*\/<\/span>\r\n  proportional_axes<\/span> =<\/span> xy<\/span>,\r\n  \/* x \u8ef8 y \u8ef8\u306e\u8868\u793a *\/<\/span>\r\n  xaxis<\/span> =<\/span> true<\/span>, yaxis<\/span> =<\/span> true<\/span>, \r\n  \/* \u8ef8\u306e\u304d\u3056\u307f\u76ee\u76db\u306e\u975e\u8868\u793a *\/<\/span>\r\n  xtics<\/span> =<\/span> false<\/span>, ytics<\/span> =<\/span> false<\/span>, \r\n\r\n  \/* \u6955\u5186*\/<\/span>\r\n  line_width<\/span> =<\/span> 1<\/span>, \r\n  color<\/span> =<\/span> red<\/span>, \r\n  parametric<\/span>(<\/span>xu<\/span>(<\/span>u<\/span>)<\/span>, yu<\/span>(<\/span>u<\/span>)<\/span>, u<\/span>, 0<\/span>, 2*<\/span>%pi<\/span>)<\/span>,\r\n\r\n\/* \u5468\u671f\u306e 1\/N \u3054\u3068\u306e\u4f4d\u7f6e *\/<\/span>\r\n  point_type<\/span> =<\/span> 7<\/span>, \r\n  point_size<\/span> =<\/span> 0.7<\/span>, \r\n  color<\/span> =<\/span> blue<\/span>,\r\n  points<\/span>(<\/span>xy_rk<\/span>)<\/span>, \r\n\r\n  \/* \u539f\u70b9 *\/<\/span>\r\n  point_type<\/span> =<\/span> 7<\/span>, \r\n  point_size<\/span> =<\/span> 1<\/span>, \r\n  color<\/span> =<\/span> black<\/span>,\r\n  points<\/span>([[<\/span>0<\/span>, 0]])<\/span>\r\n)<\/span>$\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\n\"\"<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"
\u300c\u4e07\u6709\u5f15\u529b\u306e2\u4f53\u554f\u984c\u306e\u904b\u52d5\u65b9\u7a0b\u5f0f\u3092\u6570\u5024\u7684\u306b\u89e3\u304f\u524d\u306e\u4e0b\u3054\u3057\u3089\u3048\u300d\u3067\u4e0b\u3054\u3057\u3089\u3048\u3057\u305f\u904b\u52d5\u65b9\u7a0b\u5f0f\u3092 Maxima \u3067 Runge-Kutta \u6cd5\u3092\u4f7f\u3063\u3066\u6570\u5024\u7684\u306b\u89e3\u304f\u3068\u3044\u3046\u8a71\u3002<\/p>
\u7d9a\u304d\u3092\u8aad\u3080<\/a><\/p>\n","protected":false},"author":33,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"inline_featured_image":false,"footnotes":""},"categories":[14,22],"tags":[],"class_list":["post-4245","post","type-post","status-publish","format-standard","hentry","category-maxima","category-22","nodate","item-wrap"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/posts\/4245","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/users\/33"}],"replies":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/comments?post=4245"}],"version-history":[{"count":8,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/posts\/4245\/revisions"}],"predecessor-version":[{"id":7054,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/posts\/4245\/revisions\/7054"}],"wp:attachment":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/media?parent=4245"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/categories?post=4245"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/tags?post=4245"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}